On a representation of the Verhulst logistic map

Abstract

One of the simplest polynomial recursions exhibiting chaotic behavior is the logistic map xn+1=axn(1−xn) with xn,a∈Q:xn∈[0,1]∀n∈N and a∈(0,4], the discrete-time model of the differential growth introduced by Verhulst almost two centuries ago (Verhulst, 1838) [12]. Despite the importance of this discrete map for the field of nonlinear science, explicit solutions are known only for the special cases a=2 and a=4. In this article, we propose a representation of the Verhulst logistic map in terms of a finite power series in the map’s growth parameter a and initial value x0 whose coefficients are given by the solution of a system of linear equations. Although the proposed representation cannot be viewed as a closed-form solution of the logistic map, it may help to reveal the sensitivity of the map on its initial value and, thus, could provide insights into the mathematical description of chaotic dynamics.